diff --git a/SciLean/Core/Objects/FinVec.lean b/SciLean/Core/Objects/FinVec.lean
index 094225b1..64b1ff77 100644
--- a/SciLean/Core/Objects/FinVec.lean
+++ b/SciLean/Core/Objects/FinVec.lean
@@ -12,8 +12,8 @@ The class `FinVec ι K X` guarantees that any element `x : X` can be writtens as
 ```
 -/
 class Basis (ι : outParam $ Type v) (K : outParam $ Type w)(X : Type u)  where
-  basis : ι → X
-  proj  : ι → X → K
+  basis (i : ι) : X
+  proj  (i : ι) (x : X) : K
 
 /-- Dual basis of the space `X` over the field `K` indexed by `ι` 
 
@@ -27,16 +27,16 @@ and that it is dual to the normal basis
 ```
 -/
 class DualBasis (ι  : outParam $ Type v) (K : outParam $ Type w) (X : Type u) where
-  dualBasis : ι → X
-  dualProj  : ι → X → K
+  dualBasis (i : ι) : X
+  dualProj  (i : ι) (x : X) : K
 
 /-- This should somehow relate to raising and lowering indices but I forgot how.
 
 TODO: add explanation why this is useful
 -/
 class BasisDuality (X : Type u) where
-  toDual   : X → X  -- transforms basis vectors to dual basis vectors
-  fromDual : X → X  -- transforma dual basis vectors to basis vectors
+  toDual   (x : X) : X  -- transforms basis vectors to dual basis vectors
+  fromDual (x : X) : X  -- transforma dual basis vectors to basis vectors
 
 section Basis
 
@@ -177,11 +177,6 @@ theorem proj_basis (i j : ι)
   : ℼ i (ⅇ[X] j) = if i=j then 1 else 0 :=
 by simp only [←inner_dualBasis_proj, inner_basis_dualBasis, eq_comm]; done
 
-@[simp]
-theorem proj_zero 
-  : ℼ i (0 : X) = 0 :=
-by sorry_proof
-
 @[simp]
 theorem dualProj_dualBasis (i j : ι)
   : ℼ' i (ⅇ'[X] j) = if i=j then 1 else 0 :=
@@ -211,16 +206,17 @@ instance [EnumType ι] [EnumType κ] [Zero X] [Basis κ K X] [OrthonormalBasis 
   is_orthonormal := by simp[Inner.inner, Basis.basis]; sorry_proof
 
 
-instance (priority:=high) {ι K} [EnumType ι] [IsROrC K]
+instance (priority:=high) {ι : Type} {K : Type v} [EnumType ι] [IsROrC K]
   : FinVec ι K (ι → K) where
   is_basis := sorry_proof
   duality := sorry_proof
   to_dual := sorry_proof
   from_dual := sorry_proof
 
-instance {ι κ K X} [EnumType ι] [EnumType κ] [IsROrC K] [FinVec κ K X]
+instance {ι κ : Type} {K X : Type _} [EnumType ι] [EnumType κ] [IsROrC K] [FinVec κ K X]
   : FinVec (ι×κ) K (ι → X) where
   is_basis := sorry_proof
   duality := sorry_proof
   to_dual := sorry_proof
   from_dual := sorry_proof
+
diff --git a/SciLean/Core/Objects/SemiInnerProductSpace.lean b/SciLean/Core/Objects/SemiInnerProductSpace.lean
index bb334979..8e84ebc6 100644
--- a/SciLean/Core/Objects/SemiInnerProductSpace.lean
+++ b/SciLean/Core/Objects/SemiInnerProductSpace.lean
@@ -225,6 +225,6 @@ instance (X Y) [SemiInnerProductSpace K X] [SemiInnerProductSpace K Y] : SemiInn
 
 -- instance (X) [SemiInnerProductSpace K X] (ι) [Fintype ι] : SemiInnerProductSpace K (ι → X) := SemiInnerProductSpace.mkSorryProofs
 -- instance (X) [SemiInnerProductSpace K X] (ι) [EnumType ι] : SemiInnerProductSpace K (ι → X) := SemiInnerProductSpace.mkSorryProofs
-instance (priority:=low) (ι) (X : ι → Type _) [∀ i, SemiInnerProductSpace K (X i)] [EnumType ι] : SemiInnerProductSpace K ((i : ι) → X i) 
+instance (ι) (X : ι → Type _) [∀ i, SemiInnerProductSpace K (X i)] [EnumType ι] : SemiInnerProductSpace K ((i : ι) → X i) 
   := SemiInnerProductSpace.mkSorryProofs
 
diff --git a/test/basic_gradients.lean b/test/basic_gradients.lean
index c355b966..d0e168ee 100644
--- a/test/basic_gradients.lean
+++ b/test/basic_gradients.lean
@@ -227,9 +227,11 @@ example  (w : K ^ (Idx' (-5) 5 × Idx' (-5) 5))
   : (∇ (x : K ^ (Idx 10 × Idx 10)), ⊞ (i : Idx 10 × Idx 10) => ∑ j, w[j] * x[(j.1.1 +ᵥ i.1, j.2.1 +ᵥ i.2)])
     = 
     fun _x dy => 
-      ⊞ i => ∑ j, w[j] * dy[(-j.fst.1 +ᵥ i.fst, -j.snd.1 +ᵥ i.snd)] :=
+      -- ⊞ i => ∑ j, w[j] * dy[(-j.fst.1 +ᵥ i.fst, -j.snd.1 +ᵥ i.snd)] :=
+      ⊞ i => ∑ (j : (Idx' (-5) 5 × Idx' (-5) 5)), w[(j.2,j.1)] * dy[(-j.2.1 +ᵥ i.fst, -j.1.1 +ᵥ i.snd)] :=
 by
-  conv => lhs; autodiff; autodiff; simp
+  conv => 
+    lhs; autodiff; autodiff; simp
   
   
 
diff --git a/test/generate_ftrans.lean b/test/generate_ftrans.lean
index 4c81dad4..f0bc7999 100644
--- a/test/generate_ftrans.lean
+++ b/test/generate_ftrans.lean
@@ -90,8 +90,8 @@ info: mymul.arg_xy.IsDifferentiable_rule.{w, u} {K : Type u} [instK : IsROrC K]
 
 variable
   {K : Type u} [RealScalar K]
-  {X : Type u} [SemiInnerProductSpace K X]
-  {ι : Type v} {κ : Type v'} [EnumType ι] [EnumType κ]
+  {X : Type v} [SemiInnerProductSpace K X]
+  {ι : Type} {κ : Type} [EnumType ι] [EnumType κ]
 
 set_default_scalar K
 
@@ -99,7 +99,7 @@ def matmul  (A : ι → κ → K) (x : κ → K) (i : ι) : K := ∑ j, A i j *
 
 #generate_revCDeriv matmul A x 
   prop_by unfold matmul; fprop
-  trans_by unfold matmul; autodiff; autodiff
+  trans_by unfold matmul; ftrans only; autodiff; autodiff
 
 #generate_revCDeriv matmul A x | i
   prop_by unfold matmul; fprop